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A CAT(0) cube complex is a simply connected nonpositively curved cube complex.

A cube complex, in the cubical CW complex convention used here, is built from standard hypercubes $[0,1{]}^n$ by identifying their faces of a cube using isometries. The restriction of each characteristic map to a face of a cube agrees with the characteristic map of a lower-dimensional cell, after an isometry of that face. The cubes carry their Euclidean metrics, giving an induced length distance on each connected component.^[ref]

Face identifications may identify different boundary points of the same cube. For example, one interval with its endpoints identified is a circle, and one square with opposite sides identified is a torus. The latter has one vertex, two edges and one square; it is the Salvetti complex for two commuting generators.

A stricter polyhedral convention requires every closed cube to embed and every pair of cubes to meet in one common face. That convention needs a finer cell decomposition of these examples. The cubical-CW convention allows the one-vertex quotient models used in geometric group theory. Being a nonpositively curved cube complex is an additional condition, detected by the link of a vertex in a cube complex.

For a finite graph $\Gamma$, its Right-angled Artin group isIts Salvetti complex $S_{\Gamma }$ is the one-vertex cube complex with an oriented loop labelled $v$ for each vertex of $\Gamma$, a square torus for every edge, and more generally one cubulated $k$-torus for every $k$-clique, attached compatibly along coordinate subtori. Thus ${\pi }_1(S_{\Gamma })\cong A_{\Gamma }$. The vertex link of a Salvetti complex is flag, so $S_{\Gamma }$ is a nonpositively curved cube complex.